Space charge resonances and instabilities
I thought it would be interesting to create some animations based on a few of the discussions in the book Space Charge Physics for Particle Accelerators by Ingo Hofmann. The scripts I used can be found here.
Hofmann discriminates between incoherent effects that involve a single particle and coherent effects that involve all particles in the beam. Nonlinear resonances driven by higher-order magnetic multipoles are examples of incoherent effects. Expanding the transverse magnetic field in a power series results in the addition of nonlinear terms to the equation of motion:
$$ x'' + k(s)x = \sum_{i,j=1}^{\infty}{a_{i, j}x^i y^j}. $$
The coefficients $a_{i,j}$ may be periodic — a single magnet error in a ring, for example. To avoid resonant behavior, the individual particle tunes $\nu_{0x}$ and $\nu_{0y}$ should be carefully chosen to avoid the lines defined by $M_x\nu_{0x} + M_y\nu_{0y} = N$. Space charge can have the negative effect of decreasing the tunes such that they approach these lines. This is the primary concern in circular accelerators.
It’s also possible for the beam itself to provide these higher-order terms through its electric field. Consider a matched beam — one whose density profile repeats itself after one lattice period —and suppose we track a particle through the external focusing fields and the beam’s electric field without affecting the beam (this is an approximation). The beam’s electric field may be able to be expanded in an infinite series, and the periodicity of the coefficients in this expansion will lead to additional resonance lines that the particle should avoid: $M_x\nu_{x} + M_y\nu_{y} = N$, where $\nu_x$ and $\nu_y$ are the depressed tunes. This is called an incoherent space charge resonance.
Hofmann simulates the case $4\nu_x = 4\nu_y = 360$ deg. using a PIC simulation. An evenly spaced FODO lattice is used with zero-current tunes $\nu_{0x} = \nu_{0y} = 100$ deg. in both planes. The initial beam is a Gaussian distribution truncated at three standard deviations, the emittances are the same in both planes, and the rms beam dimensions are matched to the lattice using the KV envelope equations. Hofmann uses an elliptical longitudinal distribution, but I’ll use a uniform longitudinal distribution with no energy spread (coasting beam). The beam intensity is then chosen so that the depressed phase advances are $\nu_x = \nu_y = 92$ deg., just a bit above the resonance condition.1 The simulation proceeds by slowly decreasing the zero-current phase advances: $100$ deg. $\rightarrow$ $90$ deg. over 500 cells. In this way, the core of the beam should remain approximately matched to the lattice. Below is the evolution of the horizontal phase space projection as calculated by PyORBIT.